NAME _____________________________________________ DATE ____________________________ PERIOD _____________
5-3 Study Guide and Intervention
Solving Multi-Step Inequalities
Solve Multi-Step Inequalities To solve linear inequalities involving more than one operation, undo the operations in
reverse of the order of operations, just as you would solve an equation with more than one operation.
Example 1: Solve 6x – 4 ≤ 2x + 12.
6x – 4 ≤ 2x + 12
6x – 4 – 2x ≤ 2x + 12 – 2x
4x – 4 ≤ 12
4x – 4 + 4 ≤ 12 + 4
Example 2: Solve 3a – 15 > 4 + 5a.
3a – 15 > 4 + 5a
Original inequality
3a – 15 – 5a > 4 + 5a – 5a
Subtract 2x from each side.
Simplify.
–2a – 15 + 15 > 4 + 15
Add 4 to each side.
Simplify.
–2a > 19
4𝑥
4
Divide each side by 4.
−2𝑎
−2
16
4
x≤4
Subtract 5a from each side.
–2a – 15 > 4
Simplify.
4x ≤ 16
≤
Original inequality
Add 15 to each side.
Simplify.
19
Divide each side by –2
< −2
and change > to <.
Simplify.
a<
The solution is {x । x ≤ 4}.
1
−9 2
Simplify.
1
The solution is {a । a < −9 2}
Exercises
Solve each inequality. Check your solution.
𝑞
1. 11y + 13 ≥ –1
2. 8n – 10 < 6 – 2n
3. 7 + 1 > –5
4. 6n + 12 < 8 + 8n
5. –12 – d > –12 + 4d
6. 5r – 6 > 8r – 18
8. 7.3y – 14.4 > 4.9y
9. –8m – 3 < 18 – m
7.
−3𝑥 + 6
2
≤ 12
10. –4y – 10 > 19 – 2y
11. 9n – 24n + 45 > 0
12.
4𝑥 − 2
5
≥ –4
Define a variable, write an inequality, and solve each problem. Check your solution.
13. Negative three times a number plus four is no more than the number minus eight.
14. One fourth of a number decreased by three is at least two.
15. The sum of twelve and a number is no greater than the sum of twice the number and –8.
Chapter 5
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Glencoe Algebra 1
NAME _____________________________________________ DATE ____________________________ PERIOD _____________
5-3 Study Guide and Intervention (continued)
Solving Multi-Step Inequalities
Solve Inequalities Involving the Distributive Property When solving inequalities that contain grouping symbols,
first use the Distributive Property to remove the grouping symbols. Then undo the operations in reverse of the order of
operations, just as you would solve an equation with more than one operation.
Example : Solve 3a – 2(6a – 4) > 4 – (4a + 6).
3a – 2(6a – 4) > 4 – (4a + 6)
3a – 12a + 8 > 4 – 4a – 6
Original inequality
Distributive Property
–9a + 8 > –2 – 4a
Combine like terms.
–9a + 8 + 4a > –2 – 4a + 4a
Add 4a to each side.
–5a + 8 > –2
–5a + 8 – 8 > –2 – 8
–5a > –10
a<2
Combine like terms.
Subtract 8 from each side.
Simplify.
Divide each side by –5 and change > to <.
The solution in set–builder notation is {a । a < 2}.
Exercises
Solve each inequality. Check your solution.
1. 2(t + 3) ≥ 16
2. 3(d – 2) – 2d > 16
3. 4h – 8 < 2(h – 1)
4. 6y + 10 > 8 – (y + 14)
5. 4.6(x – 3.4) > 5.1x
6. –5x – (2x + 3) ≥ 1
7. 3(2y – 4) – 2(y + 1) > 10
8. 8 – 2(b + 1) < 12 – 3b
9. –2(k – 1) > 8(1+ k)
10. 0.3( y – 2) > 0.4(1 + y)
11. m + 17 ≤ –(4m – 13)
12. 3n + 8 ≤ 2(n – 4) – 2(1 – n)
13. 2(y – 2) > –4 + 2y
14. k – 17 ≤ –(17 – k)
15. n – 4 ≤ – 3(2 + n)
Define a variable, write an inequality, and solve each problem. Check your solution.
16. Twice the sum of a number and 4 is less than 12.
17. Three times the sum of a number and six is greater than four times the number decreased by two.
18. Twice the difference of a number and four is less than the sum of the number and five.
Chapter 5
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Glencoe Algebra 1